A note on the kissing numbers of superballs
For an \(n\)-dimensional convex body \(K\subset{\mathbb R}^n\), the largest number of non-overlapping translates of \(K\) such that \(K\) has a common (boundary) point with each translate is called the translative kissing number of \(K\) and is denoted by \(N(K)\). Furthermore, for \(\alpha\geq 1\) the \(n\)-dimensional superball is defined as \({\mathcal B}_{\alpha}=\left\{(x_1,\dots,x_n)\in{\mathbb R}^n:\bigl(\sum_{i=1}^n| x_i| ^{\alpha}\bigr)^{1/\alpha}\leq 1\right\}\). In [Discrete Comput. Geom. 21, 233--242 (1999; Zbl 0924.52011)], \textit{D. E. Larman} and \textit{C. Zong} determined the translative kissing number for different polyhedra and gave an exponential lower bound for \(n\)-dimensional superballs. In the present paper the author provides an improvement of this lower bound, showing that \((1/n)\log_2N({\mathcal B}_{\alpha})\geq 1+o(1)\) as \(\alpha\) goes to infinity. Moreover, lower bounds are obtained based on binary codes which are constructive in exponential and polynomial time.
scientific article; zbMATH DE number 5146538